Adjoint-optimized metasurfaces for compact mode-division multiplexing

Optical fiber communications rely on multiplexing techniques that encode information onto various degrees of freedom of light to increase the transmission capacity of a fiber. However, the rising demand for larger data capacity is driving the need for a multiplexer for the spatial dimension of light. We introduce a mode-division multiplexer and demultiplexer design based on a metasurface cavity. This device performs, on a single surface, mode conversion and coupling to fibers without any additional optics. Converted modes have high fidelity due to the repeated interaction of light with the metasurface’s phase profile that was optimized using an inverse design technique known as adjoint analysis. We experimentally demonstrate a compact and highly integrated metasurface-based mode multiplexer that takes three single-mode fiber inputs and converts them into the first three linearly polarized spatial modes of a few-mode fiber with fidelities of up to 72% in the C-band (1530–1565 nm).


Derivation of the gradient of the figure of merit for adjoint analysis
The following is derived assuming the non-resonant physical model for better physical intuition. The final gradient in equation (S9) still applies for the cavity model except the superscript ( ) is dropped since the derivative is no longer calculated at each plane due to the steady state field assumption in the device.
Here we consider a stack of equidistantly spaced transmission masks ( ) = exp( ( ) ) , = 1 … where ( ) is the th mask. The field at different position, illustrated as the dashed lines, are ( ) , = 0 … + 1. Then (0) corresponds to the input field, and ( +1) corresponds to the output field. is the free space propagator. where is the desired target field located at plane = + 1 We can rewrite Equation (S1) as a large linear equation Our goal is then to calculate ( ) given that is an explicit function of , and satisfies the linear equation (S3).
From adjoint analysis [1], the derivative of the figure of merit with respect to the design variable can be expressed as: To take a closer look at each term Plugging these partial derivatives back, Eqn. (S4) becomes Now we define the adjoint field ≡ − † which is physically the field backpropagating from the target field. Clearly at = + 1, ( +1) = .
For some plane , ( ) = ( ) † † ( +1) which is equivalent in saying the adjoint field at some plane is equal to the field after it in space that has been backpropagated and multiplied Notice that ( † ( +1) ) * is just a complex number and serves the role to correct the phase. To paraphrase Eqn. (S7), the dependence of the objective function on the phase value on the ℎ phase plate ( ) depends only on the multiplication of the forward field ( ) and the adjoint field ( ) at the same location.
If we have multiple objectives, then we can create an overall objective based on the subobjectives = ( 1 , 2 , … , ). For example, the simplest objective function can be just the average of all objectives. Then Then using chain rule, we have Similar results can be derived for different objectives, such as = max k if we want to minimize the worst-case loss.

Alignment tolerance of fiber to the metasurface mode multiplexer (MUX)
Here we would like to quantify the alignment tolerance of the input and output fibers with respect to the metasurface device. Since our device is a mode conversion device, the additional coupling loss to the few-mode fiber (FMF) due to its misalignment can be simply calculated as the insertion loss between the ideal FMF mode and the actual FMF mode outputted by the device because they will be misaligned by the same amount. Since a mode conversion device is reciprocal in function (either with single-mode fiber (SMF) as input and FMF as output, or vice versa), the alignment tolerance on the SMF side is the same.
To further verify this, we calculate the insertion loss when the input SMF is laterally offset by some amount. Here we take the LP11b mode as an example. The result is shown below on the left (Fig. S2a). The blue and orange lines correspond to cases where the input SMF fiber is laterally offset in the or direction, respectively, and the insertion loss to the LP11b mode is simulated for the fixed output FMF. The green dashed line corresponds to directly calculating the insertion loss between two SMFs that are offset by the same amount. One can see that three curves are essentially equivalent. The same argument S4 applies to the tilt tolerance (Fig. S2b). Modeling results are shown below on the right. Thus, even though our device is highly integrated, there are no additional drawbacks in alignment for the input and output fiber.

Scalability of the metasurface mode MUX
An advantage of the cavity metasurface MUX design over existing MUX technologies is the ability to scale to a greater number of modes without significant reduction in performance and none in fabrication effort. To demonstrate this, we designed a 12-mode (6 spatial modes and 2 polarization states) MUX using the adjoint analysis technique described in the main text. Figure S3a and b show the optimized phase profile and simulated insertion loss over wavelength, respectively. Here the non-resonant design is shown due to having better bandwidth performance than a resonant design. Table S1 shows the crosstalk matrix at 1550 nm. Note that, despite scaling to twice as many modes of the fabricated design, the substrate thickness (i.e. cavity size) remains the same and the lateral size increases by an amount that can still be easily written by electron beam lithography.

Finite-difference time domain (FDTD) library of a-Si nanopillars at = 1550 nm
FDTD simulations (Ansys Lumerical Canada Ltd.) of the nanopillar library were conducted assuming Bloch boundary conditions with angles of incidence ranging from 0 o to 15 o . Note that this does not necessarily guarantee angle insensitivity. When combined to make a certain phase profile, the nanopillars as a whole can have an angle-dependent response (structural birefringence) that cannot be predicted from this simulation alone since the Bloch boundary conditions are no longer accurate. However, for low spatial frequency phase patterns the approximation is valid. Figure S5 shows that for angles below 15 o , the phase shift response is insensitive.

Measurement setup for imaging mode conversion
Measurement setup for imaging the modes coming out of the metasurface MUX (Fig. S6).
Incident light with wavelength 1550 nm from the tunable laser was fiber coupled to a single mode fiber (SMF) and precisely aligned to the input ports of the MUX. Depending on which input port was excited, a unique LP mode was emitted from the output aperture of the MUX. The image of the mode was captured by the InGaAs camera.

Measurement setup for characterizing device loss
In the first setup (Fig. S7a), the output mode from the MUX is coupled to a corresponding FMF with matching mode distribution and the coupled power is measured at the end of the FMF with a detector. This is done for each input just as in S6. The power captured from this output represents purely the light in the target mode assuming the loss in the FMF is negligible. This is defined as insertion loss in the main text. In the second setup (Fig. S7b), the total optical power coming from the output port of the MUX is measured. This is done for each input port by adding a pinhole that only allows light from the output aperture of the MUX to be captured (light exiting the boundaries of the MUX mirrors is blocked). The power captured from this output represents loss from within the device including unwanted diffraction, material absorption, as well as fabrication imperfections. This is defined as internal device loss in the main text. is negligible thus the difference between the two losses must come from the mode overlap of the converted mode by the MUX and the desired target mode of the FMF.

Polarization dependence measurement setup
The setup shown below (Fig. S8) was used to measure the extinction ratio of each mode of the metasurface mode MUX. This was done to gauge the polarization dependence of the device. In this setup there is an electric polarization controller that controls the source polarization incident on the metasurface MUX and there is a polarizer at the output side that analyzes the polarization state. For a given input polarization and mode, the power is measured when the second polarizer is co-polarized and then the power is measured again when the second polarizer is cross-polarized for the same mode.

Polarization dependence of the metasurface MUX
In order to gauge the polarization dependence of our metasurface mode MUX, the extinction ratio was measured for each mode and then FMF mode images were captured for different polarizations. The extinction ratio is defined as the power of mode with the desired polarization over the power of the mode in the unwanted polarization. Results were obtained from the setup in Fig. S8. For the FMF mode images, the polarizer and detector at the output side is replaced by the FMF and a camera in Fig. S8.

Crosstalk measurement setup
A schematic of the measurement setup used for the crosstalk of the metasurface mode MUX is presented below in Fig. S10. This method is based on Ref. 29 in the main text. A vector network analyzer (VNA) is used to control the modulation frequency of a narrow linewidth continuous-wave source via an intensity modulator (IM). The modulated light is launched into the metasurface mode MUX and the output modes are then coupled to the fiber under test (FUT) i.e. the FMF. At the end of the FMF, a photodetector (PD) is used to convert the optical signal back to an electrical signal. This is the transmission over the range of modulation frequency. Due to crosstalk of the MUX device, the light before the FMF will be a mix of the two LP mode groups. After this mix is launched into the FMF, the input pulse will separate in time based on modal dispersion. Because a narrow linewidth source is used, chromatic dispersion can be neglected and a Fourier transform of the output pulse will decompose the signal into its modal constituents weighted by the relative power in each mode. Based on the definition of crosstalk given in the main text, the crosstalk can be calculated from these weights. Note that this method only distinguishes between the LP01 and LP11 mode groups.